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Extra info for Advanced Quantum Theory and Its Applications Through Feynman Diagrams

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15) for a plane wave of momentum Pi. Interpretation of Negative-Energy States. HistoricaIly, the resolution of the negative-energy and negative-probability-density problems led to a reformulation of the Klein-Gordon theory in a many-body context. 12a) as the charge density of a positively charged, positive-energy particle "propagating" forward in time (t > 0, E > 0) via the plane-wave phase e- iEt . 12b) as the charge density of a positively charged, negative-energy state propagating backward in time (t = -Itl < 0) via eiEt = e-iEjtj.

Matrix Representation. 46) that direct-product representations such as (j, 0) + (0, j) will often occur, and it will be useful to combine them into a 2(2j + 1)-dimensional irreducible representation corresponding to the matrix ~ Ul(A) _ (DUl(A) 0 0) DUl(A) . 45) along with ß = ß- 1 = (~ ~). 46) is further reinforced by the space reflection operation (to be discussed in Chapter 6) under which DU) -+ DU). 46) are the finite, nonunitary irreducible representations of the larger group 2', inc1uding the velocity transformations, rotations, and also space, time, and space-time reflections.

1Ieft-handed state survives for the (0, 1) representation in the massless limit. 1. = ±1, massless spin-! , neutrinos, antineutrinos) must have their momenta lined up along or against their spin, with only one configuration allowed for each irreducible representation of 2. We shall conc1ude later that nature chooses the (0,1) representation for the (left-handed) neutrino and the H, 0) representation for the (right-handed) antineutrino. Spin-l Helicity States. 1. 1. 1. = 0 is a longitudinal state polarized in the z-direction for massive spin-1 partic1es.